Colin McLarty, "Theology and its discontents: The origin myth of modern mathematics," http://people.math.jussieu.fr/~harris/theology.pdf . Debunks the myth that Gordan denounced Hilbert's proof of the basis theorem with the dismissive sentence, "This is not mathematics; this is theology!"

Some might say that my question belongs on the Historia Matematica mailing list; however, besides the fact that I don't subscribe to Historia Matematica, I think that the superior infrastructure of MathOverflow actually makes it a better home for the list I hope to create. Still, maybe someone should let the Historia Matematica mailing list know that I'm asking the question here.

$\begingroup$Let's hope John Stillwell checks in at some point on this question:He,of course,has a reputation as one of mathematics' most knowledgeable and prolific historical scholars.$\endgroup$
– The MathemagicianMay 31 '10 at 20:10

$\begingroup$It's difficult not to get confused here. Both Euclids look the same to me. Both of them have long beards, they both are holding a compass, etc. In the Legendre case the situation was easier: the fake Legendre had a wig whereas the real one did not.$\endgroup$
– José Hdz. Stgo.Jun 1 '10 at 7:03

9

$\begingroup$It's easy: fake Euclid is wearing a bandana! Also, EVCLIDI MEGAREN should be a bit of a clue.$\endgroup$
– Victor ProtsakJun 1 '10 at 7:32

9

$\begingroup$Great. Apparently medieval artists had a similar lack of clue about ancient Greece as the ones of today have about the Middle Ages. ;)$\endgroup$
– darij grinbergJun 1 '10 at 11:51

There is a nice article by Brian Hayes in the American Scientist (May-June 2006), "Gauss' Day of Reckoning," in which he looks at the history of the so-called "baby Gauss" story (that Gauss amazes his teacher by summing the first 100 positive integers).

There is a famous anecdote about Euler embarrassing Diderot in Catherine the Great's court. He claimed to have mathematical proof of the existence of God, when in fact he just stated mathematical nonsense (which Diderot did not understand): "Monseur, $(a+b^n)/n=x$ donc Dieu existe; répondez!" B. H. Brown tracked down the source of this myth in "The Euler-Diderot Anecdote" (Amer. Math. Monthly, Vol 49, 1942, reprinted in William Dunham's The genius of Euler: reflections on his life and work (2007))

Finally, there is a mathematical urban legend that I thought was surely was false, but is apparently true (according to Snopes). This is the story about the student who comes late to class and sees the homework written on the board. After a lot of effort he solves the problems. Only later did he discover that they were not homework, but open problems. It turns out that the student in the story was George Dantzig. Snopes cites a 1986 interview with Dantzig from the College Mathematics Journal.

$\begingroup$Dantzig also recounts the story in his interview in More Mathematical People. He adds that Knuth had hailed him and said he (Dantzig) was now an influence on Christians in mid-west America; apparently the minister of the church (Norman Vincent Peale, perhaps?) was seated next to Dantzig on a plane and was explaining to him his views on positive thinking, and Dantzig had offered this story which the minister later used to illustrate his thesis in his sermon.$\endgroup$
– Todd Trimble♦Dec 31 '15 at 13:33

Historians like nothing better than to debunk other historians,
so there are plenty of papers shooting down one myth or another.
I enjoy these papers as much as the next person, but sometimes the
debunking becomes the new myth. So, if I may turn the question
round a bit, here is a case where I think the original "myth" has
merit, and I'm not convinced by the "debunking".

"Myth". The ancient Mesopotamians who compiled Plimpton 322 were
looking for Pythagorean triples and had a powerful method for
finding them. For example, they found the triple (13500,12709,18541).

"Debunking". By Eleanor Robson in Historia Mathematica 28 (2001) 167-206,
which may be viewed here.

$\begingroup$I would like to know whether there are rejoinders to Rothman's article mentioned above. That Bell's Men of Mathematics is unreliable goes without saying. However, there are several things to object to in Rothman's article. It seems to have been written with an intense dislike for romantic sensibilities - or even republican sympathies - that passes itself off for a tone of objectivity. For Rothman, Galois's troubles - including being expelled for exposing the opportunism of his school's director - were "what might have been anticipated", and Galois's objections to this are "paranoia".$\endgroup$
– H A HelfgottJun 7 '10 at 11:45

$\begingroup$Not exactly a rejoinder to Rothman, but a nice attempt to tell the Galois story from a fresh point of view, is the novel The French Mathematician by Tom Petsinis (Walker & Company 1998).$\endgroup$
– John StillwellJun 7 '10 at 12:15

$\begingroup$@H A Helfgott: Such a rejoinder to Robson (not Rothman) was published by Britton, Proust, and Shnideer here. See also mathscinet review.$\endgroup$
– Mikhail KatzJul 25 '13 at 12:34

$\begingroup$@Mikhail Katz: Thanks. I found Robson's article to be written in an overly polemical fashion. Indeed, she acknowledges it is written polemically - the problem is that it gives the impression of debunking more than it does. It does do a good job of casting doubt (or even scorn) on speculation about applications - but it gives the impression of casting down on the non-myth above, while grudgingly acknowledging the grounds for it towards the end.$\endgroup$
– H A HelfgottOct 4 '17 at 19:46

The golden ratio, also called by different authors the golden section [Cox], golden number [Fi4], golden mean [Lin], divine proportion [Hun], and division in extreme and mean ratios [Smi], has captured the popular imagination and is discussed in many books and articles. Generally, its mathematical properties are correctly stated, but much of what is presented about it in art, architecture, literature, and esthetics is false or seriously misleading. Unfortunately, these statements about the golden ratio have achieved the status of common knowledge and are widely repeated. Even current high school geometry textbooks such as [Ser] make many incorrect statements about the golden ratio.

$\begingroup$I vaguely remember reading a review of a more recent book that debunked many of the myths (e.g. by empirically measuring the most pleasing ratio of sides of rectangular objects).$\endgroup$
– Victor ProtsakJun 3 '10 at 1:19

2

$\begingroup$You might be thinking of Mario Livio's book; there's a chapter where he does that.$\endgroup$
– Qiaochu YuanJun 3 '10 at 2:13

One of the biggest myths in number theory is that work on Fermat's last theorem played a large role in the development of ideal theory and algebraic number theory. In fact it was much loftier goals such as the quest for higher reciprocity laws that were the true sources of inspiration. For references see e.g. Lemmermeyer's book "Reciprocity Laws" p. 15 (notes on Lagrange).

Speaking of FLT, recall the legend that Kummer submitted a false proof based on the erroneous assumption that cyclotomic number rings were UFDs. Edwards argued that this may be a myth and put forth an argument that, instead, Kummer's mistake was based on a simple error not related to any UFD assumption. However R. Bolling recently discovered new evidence that seems to lend strong support to the veracity to the claim that Kummer did in fact mistakenly assume facts equivalent to unique factorization in various rings of cylotomic integers in one of his early papers (which was not an attempted proof of FLT). So only part of this legend is actually true.

The above two paragraphs don't really do justice to the complex history. For a more faithful rendition I highly recommend that the interested reader also consult Franz Lemmermeyer's recent paper [1] which, imho, is one of the most interesting historical works on number theory in quite some time.

$\begingroup$As for the FLT legend, Boelling's work shows clearly that Kummer's erroneous manuscript had nothing to do with FLT. The topic of Kummer's research was the "splitting of primes" in cyclotomic rings, and his error is related with unique factorization. $\endgroup$
– Franz LemmermeyerJul 2 '10 at 6:18

7

$\begingroup$This is the first time I have ever heard a story in which Kummer made a mistake of assuming unique factorization. In the story I have always heard, it was Lame who made this error, and Kummer who pointed it out. See for example bookrags.com/research/fermats-last-theorem-wom$\endgroup$
– Timothy ChowJul 2 '10 at 15:52

$\begingroup$Oops, too much cutiing and pasting in the middle of the night. Of course I didn't mean to say Kummer's false proof was on FLT (like Lame's). I corrected that. Thanks for the heads-up. $\endgroup$
– Bill DubuqueJul 2 '10 at 18:19

3

$\begingroup$Kummer did not assume unique factorization - he "proved" a theorem showing that the prime ideals above prime p = 1 mod q are principal. This implies, by what he would prove later (the class group is generated by ideals of degree 1) that the ring of integers in this cyclotomic field has uique factorization. As for Lame's error: this was pointed out by Liouville - Kummer explicitly constructed a counterexample (23rd roots of unity) which (my guess) he learned from Jacobi. Jacobi probably didn't think for a second that this example might make history . . . $\endgroup$
– Franz LemmermeyerJul 3 '10 at 16:52

4

$\begingroup$Timothy, is it possible you never read Bell's Men Of Mathematics? Here's the Kummer story/legend/myth, from p.473: Kummer too had fallen afoul of the net which snared Cauchy, and for a time he believed that he had proved Fermat's "Last Theorem." Then Dirichlet, to whom the supposed proof was submitted for criticism, pointed out by means of an example that the fundamental theorem of arithmetic, contrary to Kummer's tacit assumption, does {\it not} hold in the field concerned. $\endgroup$
– Gerry MyersonJul 5 '10 at 0:45

I like this question very much - many popular history of mathematics books seem filled with legends more than history.

I would expand the question "What are some good papers that debunk common myths...?" to include books as well as papers. Maybe the best books to debunk myths are books that directly discuss and include the primary source material (in translation, perhaps). A fantastic recent book in this spirit is "The Mathematics of Egypt, China, India, and Islam: A Sourcebook," edited by Victor J. Katz. This is a great value as well! For example, an scholarly translation of the Chinese "Nine Chapters on the Mathematical Art" costs around 350 dollars on Amazon -- but one can instead find it in this sourcebook, together with a multitude of other translated texts, for around 50 or 60 dollars.

To demonstrate that this book addresses your specific question, on pages 467-477 you can find a translation of the Bijaganita of Bhaskara II. At the end is Verse 129, the end of which is translated, "Hence, for the sake of brevity, the square-root of the sum of the squares of the arm and upright is the hypotenuse: thus it is demonstrated. And otherwise, when one has set down those parts of the figure there, [merely] seeing [it is sufficient].

The author of this section then mentions "These verses are presumably the ultimate source of the widespread legend that Bhaskara gave a proof of the Pythagorean theorem containing only the square figure shown in figure 4.19 and the word 'Behold!' "

The figure (4.19 in the book) is not among the verses in an old text, as far as I know. I think that Indian texts of the period were traditionally written on palm leaves (which degrade somewhat quickly), and copied every generation or two, so we don't have very old texts. In any case, the "Behold!" legend for the Pythagorean theorem seems to be a myth or at least a vigorous embellishment of history.

There is a rumor (a myth?) in holomorphic dynamics about a rivalry between Fatou and Julia competing for the 1918 Grand Prize of the french science academy. In fact Fatou never submitted a memoir for the Prize.

What we now call the Julia set was first defined in a note in Comptes Rendus by Montel, and is the starting point for the work of Fatou and Julia. Fatou published a few notes on the subject in the Comptes Rendus before Julia, who started a quarrel, claiming priority for the results. Fatou didn't try to fight back, apparently because he didn't bother, and the Academia fullfilled the wish of Julia. This may be the source of the rumor.

You may also know the Parseval formula about the relationship between the square of the L2 norm of a function and its Fourier coefficients. This formula is due to Fatou. H. Lebesgue did the case of a bounded function, and Fatou extended it to any square integrable functions. This made a big impression on Lebesgue.

This is exposed in a recent book by Michele Audin entitled "Fatou, Julia, Montel (...)"

$\begingroup$What's the exact reference for Montel's paper where he introduced the Julia set ?$\endgroup$
– Alexandre EremenkoJan 13 '16 at 4:53

$\begingroup$@Eremenko Points in the Julia set are called irregular points by Montel. From Audin's book (VI.3), they appear in the 1903 and 1904 CRAS Montel's notes and in its thesis. They also appear in a 1906 CRAS note of Fatou.$\endgroup$
– coudyJun 29 '16 at 20:29

$\begingroup$Montel calls irregular the points where SOME family of functions is not normal. He did not consider families of iterates before Fatou.$\endgroup$
– Alexandre EremenkoJun 30 '16 at 16:49

I hope someone will follow up on Cooke's "Modest Proposal" (see page 473): ``I would like to invite some ambitious mathematician with time on his/her hands to write a monograph on the role played by legends in the mathematical community. Or perhaps someone would be willing to set up the mathematical equivalent of the snopes.com website, where mathematical “urban legends” can be checked out.''

$\begingroup$Cooke's paper mentions several other papers of the type I'm interested in. I'll mention just one: "The So-Called Euler-Diderot Incident" by R. J. Gillings, Amer. Math. Monthly 61 (1954), 77-80. Gillings traces the legend that Euler confounded Diderot with a nonsensical mathematical proof of the existence of God to the memoirs of Dieudonne Thiebault. Thiebault does tell the story, but in his account it's a Russian philosopher (not Euler), and Diderot knew it was nonsense but kept quiet because he sensed they were having him on. Thiebault also says he's not sure the story was true. $\endgroup$
– Timothy ChowJun 2 '10 at 15:56

"Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, pages 44‒52, by me and Catherine Woodgold, debunks the widespread belief that Euclid's proof of the infinitude of primes is a proof by contradiction. The proof that Euclid actually wrote is simpler and better than the proof by contradiction often attributed to him.

$\begingroup$A similar mistaken belief is that Cantor's diagonal argument is a proof by contradiction.$\endgroup$
– John StillwellJun 2 '10 at 23:13

12

$\begingroup$Just as Euclid proves "given finitely many primes, you can find another one," Cantor proves "given countably many real numbers, you can find another one." Also, just as Euclid's new prime can be constructed from the given primes, Cantor's new real can be constructed from the given reals.$\endgroup$
– John StillwellJun 3 '10 at 2:05

3

$\begingroup$I agree that there are constructive aspects in both arguments. But the key point is what are "given" reals. Cantor's proof starts with $\textit{assuming}$ that a bijection $f:\mathbb{N}\to\mathbb{R}$ exists and specifying or fixing it. The conclusion is that there is a number not in the image of $f$, which is a contradiction. (This is quantified over all putative bijections.) This seems a quintessential proof by contradiction to me. What do you make of Liouville's proof that transcendent numbers exist? $\endgroup$
– Victor ProtsakJun 3 '10 at 5:03

8

$\begingroup$Victor, Cantor's proof may be phrased as by contradiction, but that is not the most elegant nor even the most natural way. You can present it as follows. Given any function f:N-->R, the well-known diagonal procedure produces a real number not in the range. Therefore no function from N to R is surjective. This is not a proof by contradiction, any more than the proof that there is no largest natural number: given any n, n+1 is larger. You do not need to "assume n is the largest".$\endgroup$
– Pietro KCJun 3 '10 at 6:40

Was Cantor Surprised? published in Monthly is debunking (or trying to do so) that Cantor was so surprised when he discovered $I=[0,1]$ and $I^2$ have the same cardinality
that he said “I see it, but I don’t believe it!”. The abstract of the paper reads as follows:

Abstract. We look at the circumstances and context of Cantor’s famous remark, “I see it, but I don’t believe it.” We argue that,
rather than denoting astonishment at his result, the remark pointed to
Cantor’s worry about the correctness of his proof.

There have been recent articles in the Mathematical Intelligencer about the well-known rivalry/animosity between Erdos and Selberg. Didier adds this link, pointing to an article of Graham and Spencer that appeared in the M.I. Emerton adds this link, pointing to an article by Goldfeld on the controversy.

$\begingroup$Legend has it that the two were planning a joint publication when Selberg overheard someone say something like, "Hey, have you heard that Erdos and somebody-or-other have an elementary proof of the Prime Number Theorem?" The legend continues, that Selberg, not wishing to go down in history as "somebody-or-other," rushed the proof into print without waiting for Erdos, got all the credit, the Fields medal, etc. I haven't seen the Intelligencer and don't know what, if anything, it has to say about the legend. $\endgroup$
– Gerry MyersonJun 1 '10 at 13:04

$\begingroup$One should also consider the reference math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf, written by D. Goldfeld, which contains quotes from Selberg disputing the account of Straus that is given in the reference mentioned by Didier Piau.$\endgroup$
– EmertonJun 2 '10 at 5:43

1

$\begingroup$I had not heard of the legend mentioned by Gerry, but it seems incredibly biased. $\endgroup$
– H A HelfgottJun 2 '10 at 19:44

David Fowler's book "The Mathematics Of Plato's Academy: A New Reconstruction" sets out to deconstruct the myth that "the early Pythagoreans based their mathematics on commensurable magnitudes, but their discovery of the phenomenon of incommensurability (the irrationality of the square root of 2) showed that this was inadequate; this provoked problems in the foundation of mathematics that were not resolved before the discovery of the proportion theory that we find in Book V of Euclid's Elements". The arguments and conclusions are complex and interesting; here is a review with some of the main points: http://www.maa.org/publications/maa-reviews/the-mathematics-of-platos-academy-a-new-reconstruction.

The main thread of their revisionist picture is that 17th- and 18th-century analysis had foundations that were far more rigorous and well-defined than most modern mathematicians have been led to believe.

I would expand the question "What are some good papers that debunk common myths...?" to include web pages as well as papers. Naturally, there is a couple of candidate web pages in the reference section of the note mentioned by sigfpe. Another candidate might be http://www.snopes.com/science/nobel.asp where the myth of the rivalry between A. Nobel and Mittag-Leffler is debunked (to some extent).

$\begingroup$Snopes.com is a great website and I thought carefully about whether to phrase my question to include it or not. I decided to focus on papers because on average, people take more care with their research when publishing a paper than when posting something on a website. Snopes.com may be an exception because they take great pains to be accurate. However, often when I see a website that purports to debunk a myth, I'm not sure whether to believe it. If there are exceptionally good websites then people should mention them, but be aware that I put a very high premium on reliability.$\endgroup$
– Timothy ChowMay 31 '10 at 23:27

$\begingroup$Interesting paper...thanks! But does it really say what you claim it says? I quote: "Frechet thought of probability as an application of mathematics, not as a branch of pure mathematics itself, so he did not think he was axiomatizing probability theory. It was Kolmogorov who first called Frechet’s theory a foundation for probability theory." Looks pretty close to crediting Kolmogorov with the idea of using measure theory as a foundation for probability theory.$\endgroup$
– Timothy ChowJun 1 '10 at 15:36

1

$\begingroup$I think a common view was that Kolmogorov provided an incomplete framework for probability. Frequentists, Subjectivists etc. would provide different axiomatizations giving rise to probabilities, based on something more empirical. $\endgroup$
– Michael GreineckerJun 1 '10 at 20:44

$\begingroup$@Ilya Am I criticising you if I say that you did not invent the calculus?$\endgroup$
– Michael GreineckerDec 5 '14 at 8:30

$\begingroup$@MichaelGreinecker: certainly depends on the tone. you should have taken my original comment with a grain of salty joke :) I'm going to ask a question regarding the history of probability and measure theory on MSE in 5 minutes, may be of your interest.$\endgroup$
– IlyaDec 5 '14 at 8:31

Historians of mathematics get very cross if you dare to say "Pythagoras's theorem." You have to say, "The Pythagorean theorem," to emphasize that it wasn't the brainwave of a man called Pythagoras. Indeed, that fact, as well as the irrationality of the square root of 2, were (I read) probably known long before Pythagoras, and the only Greek proofs we know of came much later. As for the story that the penalty for revealing the irrationality of the square root of 2 to outsiders was death ...

$\begingroup$How can you know the irrationality of the square root of 2 without having a proof? Most sources I know place the first proofs after 450 BC, long after Pythagoras had died.$\endgroup$
– Franz LemmermeyerJun 1 '10 at 13:35

To respond to Timothy Chow's comment, there is a famous essay by historian Judith Grabiner entitled "who gave you the epsilon?" with the implied answer being "Cauchy". This can be viewed at http://www.ams.org/mathscinet-getitem?mr=691368

It should be noted that not all historians agree with Grabiner. Thus, Schubring in a recent text comments: "I am criticizing historiographical approaches like that of Judith Grabiner where one sees epsilon-delta already realized in Cauchy". This can be viewed at http://link.springer.com/article/10.1007/s10699-015-9424-0

$\begingroup$Interesting. I have never heard this myth before. I have always heard epsilons and deltas attributed to Weierstrass.$\endgroup$
– Timothy ChowDec 30 '15 at 21:10

$\begingroup$@TimothyChow, I added some sources to my answer to respond to your query. Let me know if you would like additional details.$\endgroup$
– Mikhail KatzDec 31 '15 at 8:52

$\begingroup$About Cantor's surprise, I read once he did write in French to Dedekind "je le vois, mais je ne le crois pas...". I don't know whether this is a myth or not.$\endgroup$
– Sylvain JULIENOct 24 '16 at 20:27

$\begingroup$@SylvainJULIEN, I am not sure why you left this comment here. I don't seem to have written anything about Cantor in my answer here, though I did in other answers. At any rate, a comment of this sort by Cantor seems to be on record. The only question is, how does one interpret it. There have been dramatically different interpretations. Why are you interested in this?$\endgroup$
– Mikhail KatzOct 25 '16 at 7:41

$\begingroup$I got mistaken, I commented above the answer about Cantor's surprise instead of below. I'm interested in this, because I'd like to know why Cantor wrote in French. How was French considered in Germany at that time ?$\endgroup$
– Sylvain JULIENOct 25 '16 at 8:31

I protest first of all against the use of an infinite quantity as a completed one, which is never permissible in mathematics. The infinite is only a façon de parler, where one is really speaking of limits to which certain ratios come as close as one likes while others are allowed to grow without restriction.

This statement is often used to claim that Gauss was opposed to the use of infinite sets. In the paper "Gauss on Infinity", (Historia Mathematica 6 (1979), 430-436), W. C. Waterhouse shows that Gauss was actually talking about the sloppy use of "circles of infinite radius" in a fallacious geometric proof by H. Schumacher of the parallel postulate. Infinite sets do not even come up, but rather it is the imprecise use of what we would now formalize as "unlimited non-standard real numbers" that Gauss is objecting to.

The article goes on to show that, curiously, it seems that it was Lipschitz, when writing to Cantor, who first interpreted the above statement by Gauss as criticizing the use of infinite sets.

I will now add my own statement going beyond Waterhouse's paper. It seems that Gauss's full criticism of Schumacher's proof, not just the excerpt commonly used, is saying that non-Euclidean planes have non-zero curvature, and it seems likely to me that if done with non-standard analysis, that is what Schumacher's proof would show, once the fallacious step was removed.

A recent one could be Peter Milne's "On Gödel Sentences and What They Say" Philosophia Mathematica (III) 15 (2007), 193–226. doi:10.1093/philmat/nkm015 debunking the myth that Gödel sentences are true because they say of themselves that they are unprovable.

Sorry, I realize this is not exactly what was being asked. However, there is still some analogy with the pattern "someone was believed to do or say something while in fact he/she didn't".

$\begingroup$You're right; this is not quite what I asked for. Misconceptions about the content of mathematical theorems are not the same as myths in the history of mathematics. As far as Goedel is concerned, my favorite expose of nonsensical interpretations of Goedel's theorem is Torkel Franzen's book "Goedel's Theorem: An Incomplete Guide to Its Use and Abuse." By the way, if you read Milne's paper carefully, you'll see that your description of its contents is not quite accurate.$\endgroup$
– Timothy ChowJun 1 '10 at 15:28

This myth is in living memory and I have not seen it debunked in print. The myth is that the Appel-Haken proof of the four colour theorem was controversial when it was published because it was a computer proof. This makes a good story but is it true?

$\begingroup$FWIW, R.Diestel writes in Graph Theory that the original approach of Appel-Haken "has not been immune to criticism, not only because of their use of a computer".$\endgroup$
– OlivierJun 1 '10 at 13:31

2

$\begingroup$This is certainly true. Nobody needs to write an article "debunking" this one because philosophers have written entire journal articles questioning the status of the theorem. See for example Thomas Tymoczko, "The four-color problem and its philosophical significance," J. Phil. 76 (1979), 57-83.$\endgroup$
– Timothy ChowJun 1 '10 at 15:13

2

$\begingroup$Well, you didn't say "mathematicians" in your original statement, and you only said "controversial," not "did not accept." But certainly, over the years, I have personally encountered several mathematicians who have expressed dissatisfaction with Appel, Haken, and Koch's use of a computer in the proof. They wanted a humanly-comprehensible, elegant proof. Now, to say that the mathematical community as a whole rejected the proof is certainly false, but I don't recall anyone ever claiming this. Maybe it's a myth that there ever was such a myth?$\endgroup$
– Timothy ChowJun 1 '10 at 16:54

4

$\begingroup$I certainly didn't accept the Appel-Haken proof (until Gonthier did a fully verified machine-checked version), for the same reasons I don't (yet) accept the Hales's proof of the Kepler conjecture. If you run a computer program as part of the proof, then you also have to supply a correctness proof of the actual computer program you used, before you really have a full proof. (And indeed, Hales and his collaborators are currently developing a fully machine-checked version.) $\endgroup$
– Neel KrishnaswamiJun 1 '10 at 22:40

Anellis, I.H. (1995), “Peirce Rustled, Russell Pierced : How Charles Peirce and Bertrand Russell Viewed Each Other's Work in Logic, and an Assessment of Russell's Accuracy and Role in the Historiography of Logic”, Modern Logic 5, 270–328. Online.